To me an abstract idea used to be a normal thought processed and re-worked until it was unrecognizable. However, I know now that it is in fact more like a seed, inspired to grow from the deepest, most inaccessible part of the unconscious.
It is what is known as a "visualization", or imagery in "the mind's eye". These acts of abstraction are impossible to communicate to others without the vehicle of media.
As intangible as it may be, an abstraction need not lose its ambiguity once it is fully conceived and produced in the form of "art". This form is simply a translation of the artist's idea represented for the purpose of observation. However, no matter how accurately executed, or how apporpriate the choice of materials may be, the work of art will never be exactly as the abstract thought.
Abstraction occurs in time. It is intangible and subjective.
My own experiences with abstraction began as early as I recognized the similarity between aural structures and visual images. This link is a completely abstract idea; there is no literal connection bewteen music and art besides written musical notation being visual in nature.
Many people argue that jazz is closest to the abstract expressionist movement in terms of style.